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Wall shear stress

The wall shear stress in single-phase flow is modeled by the standard expression

$$\tau_w = f \frac{\mbox{$<\!{G}\!>$}^2}{8 \rho_\ell}$$ (6.69)

where f denotes the Darcy-Weisbach friction factor^6.9 [--].

In diabatic two-phase flow the wall shear stress is considerably larger compared to the single-phase flow with the same mass flux. This increased shear is modeled by a two-phase friction multiplier, $\phi_{\ell 0}^2$, in the following way

$$\tau_w = f_{\ell 0} \phi_{\ell 0}^2 \frac{\mbox{$<\!{G}\!>$}^2}{8 \rho_\ell}$$ (6.70)

where the index $\ell 0$ denotes that the quantity is evaluated for a flow with liquid at the total mass flux $\mbox{$<\!{G}\!>$}$. According to (6.80) we need constitutive relations for the single-phase friction factor, $f_{\ell 0}$ and the two-phase friction factor, $\phi_{\ell 0}^2$.

The Darcy-Weisbach single-phase friction factor, f, is often evaluated by the Colebrook interpolation formula [28, p. 428]^6.10

$$\frac{1}{\sqrt{f}} \approx - 2.0 \log \left[ \frac{\epsilon / D_e}{3.7} + \frac{2.51}{\mbox{\bf Re}_{D_e} \sqrt{f}} \right]$$ (6.71)

where

>

is the Darcy-Weisbach friction factor [--].

>

is the equivalent roughness^6.11 of the wetted channel surface [m].

>

is the hydraulic equivalent diameter of the channel [m].

>

is the Reynolds number with the characteristic length equal the hydraulic equivalent diameter [--].

The hydraulic equivalent diameter concept enables us to use correlations originally derived for circular pipes. The hydraulic equivalent diameter, D_e in [m], is defined by

$$D_e \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\frac{4 A_c}{P_w}$$ (6.72)

where A_c is the cross-sectional flow area $[{\mbox{m}}^2]$ and P_w is the wetted perimeter [m].

When we wish to calculate $f_{\ell 0}$ we have to use a Reynolds number, $Re_{D_e,\ell 0}$, defined by

$$Re_{D_e,\ell 0} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$... ...iptstyle\triangle$}}\;\frac{ \mbox{$<\!{G}\!>$} D_e}{\mu_\ell}$$ (6.73)

ie where we assume a liquid flow with the total mass flux $\mbox{$<\!{G}\!>$}$.

Strictly speaking the Colebrook formula was developed for circular pipes and not rod bundles but (6.81) gives reasonably accurate results for rod bundles as well. In order to obtain more accurate expressions for the single-phase friction factor for rod bundles Rehme [38, p. 229-233] has constructed new correlations specifically for rod bundles. Because the very complicated structure of Rehme's correlations we will, however, adopt the Colebrook formula for the single-phase friction factor calculation.

Furthermore (6.81) was developed for fully developed, isothermal flow. However according to Weismann [11] the total single-phase frictional pressure drop calculated by (6.81) is accurate if the total channel length, L_c, fulfills

$$\frac{L_c}{D_e} > 100$$ (6.74)

This requirement is in general fulfilled for the BWR fuel element.

In a BWR core the flow is far from isothermal and a thermal boundary layer exists in which the liquid close to the wall has a higher temperature than the liquid further away from the wall. This phenomenon implies, due to the decreasing liquid viscosity of water with temperature, that the isothermal friction factor calculated at bulk temperature is higher than the actual friction factor. Many corrections to the isothermal friction factor exist in the literature--see, for instance, [11] and [38]. We will, however, in our analysis disregard the effects of the thermal boundary layer altogether. With this simplification it is possible to uncouple the heat transfer and hydraulics calculations.

The most famous correlation of the two-phase friction multiplier is that of Martinelli and Nelson [39]. The Martinelli-Nelson friction multiplier does not include a flow effect, ie a dependence on $\mbox{$<\!{G}\!>$}$. It has been verified by experiments that the two-phase friction multiplier, however, depends on the mass flux. In order to account for this flow effect Jones has modified the Martinelli-Nelson multiplier to include a flow effect. He proposed the following expression for $\phi_{\ell 0}^2$ [18, p. 230]

$$\phi_{\ell 0}^2 = \Psi(\mbox{$<\!{G}\!>$},p) \left\{ 1.2\l... ...f}{\rho_g} -1\right) \mbox{$<\!{x}\!>$}^{0.824} \right\} + 1.0$$ (6.75)

where the function $\Psi(\mbox{$<\!{G}\!>$},p)$ is defined by

$$\Psi(\mbox{$<\!{G}\!>$},p) \;\hbox{$=$\kern-0.68em\raise1.1... ... \mbox{$<\!{G}\!>$} \ge 0.7\cdot10^6 \\ \end{array} \right.$$ (6.76)

Note that (6.86) is a dimensional equation and assumes use of the US units, ie p is inserted in [psia] and $\mbox{$<\!{G}\!>$}$ in [ ${\mbox{lbm}}/({\mbox{h}}\cdot{\mbox{ft}}^2)$].

In the literature we find a number of other correlation of $\phi_{\ell 0}^2$, for instance those due to Baroczy [18], Thom [18] or Friedel [40]. Actually Whalley [40, p. 58] does not recommend the Martinelli-Nelson correlation (which is the foundation of the Jones correlation)--he recommends the Friedel correlation as the best available correlation (at least for $(\mu_\ell/\mu_g) < 1000$). We will, however, stick to the Jones correlation since it has a wide acceptance in the nuclear industry.

The frictional pressure drop is in general correlated against experimental data in the following way

.

Measure the total pressure drop as a function of different flow variables.

.

Calculate the frictional pressure drop from the total pressure drop by subtracting the accelerational and gravitational pressure drops for a flow model assumption, eg the homogeneous or separated model.

That is, the resulting correlation of the wall shear stress is actually dependent on the flow model which is used for evaluation of the accelerational and elevational pressure drops. Consequently, when we apply a correlation we should in fact ensure a correspondence between the flow model used and the model assumption implicit in the correlation--this restriction is, however, in practice often disregarded (or overlooked). In the case with the Martinelli-Nelson correlation [39] it is based on data in which both the gravitational and acceleration pressure changes are negligible, ie it can be used in connection with every model.

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